Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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1answer
27 views

Poincaré-Dulac for vector field

In the above calculation of $w(z)$, why would a term containing $\frac{\partial}{\partial z_j}$ appear? Since all of the substitutions in the formula don't have anything containing derivative's.
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24 views

Vector field tangent to a singular holomorphic foliation

Is there an intuitive way to see that the vector field $$\begin{equation} v=\begin{cases} \dot z=e^{\frac{1}{z}}\\ \dot w=e^{\frac{1}{w}} \end{cases} \end{equation}$$ is tangent to a singular ...
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16 views

Complex dynamics: does equicontinuity at a point imply equicontinuity in a neighborhood?

In Alan F. Beardon's book "Iteration of Rational Functions", the author defines the Fatou set $F(R)$ of a rational function $R$ as the maximal open subset of the Riemann Sphere (endowed with the ...
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1answer
50 views

Naming bulbs on the Mandelbrot set

Can anyone point me to an article or webiste that explains exactly how the bulbs of the Mandelbrot set are named. I know there are bulbs that have the names "p/q" for every set of co-prime integers. ...
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25 views

Bounding a complex iteration through the logarithmic conjugation

I am going through a paper and the following is stated as obvious. Let $f$ be a holomorphic function and $H = \{x + iy \in \mathbb{C} \;|\; x < log(\epsilon) \}$, where $0 < \epsilon < 1/2$. ...
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2answers
51 views

Hartog's Theorem and Entire Functions

I'm interested in multivariable Complex Analysis, and I have two questions: My first question is as follows: after reading about Hartog's Extension Theorem I started wondering about the following ...
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13 views

Vector field tangent to the graphs of solutions of an equation

$\dot{z}=\frac{A(t)}{t^{k+1}}$, $z\in \mathbb{C}^n$, $t\in D_1={|t|<1}, k\in \mathbb{N}(k\geq 1)$, $A(t): D_1 \to \text{End}(\mathbb{C}^n)$ is holomorphic. How to see that the following vector ...
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1answer
35 views

Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set [closed]

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that ...
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2answers
112 views

Confusing pictures about tetration !? [closed]

On the webpage http://tetration.org/Tetration/index.html, We are supposed to get an explanation of tetration, whatever that means exactly. In particular I feel the pictures are not well explained. ...
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1answer
74 views

Is the basin of attraction of a $p$ starshaped wrt to $p$?

Let $p\in\Bbb C^n$ (with $n\ge2$) be an attractive fixed point for $F\in\operatorname{Aut}\Bbb C^n$ (i.e. holomorphic bijection), that is $F(p)=p$ and all the eigenvalues of $F'(p)$ are in modulus $&...
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34 views

Complex dynamics for non-holomorphic functions

Are there any resources for studying the dynamics of functions that are not holomorphic? Specifically, I am trying to study (for a paper) the behaviour of the function $$N(z) := z - \frac{2\overline{...
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3answers
103 views

Why the cardioid shows up in the Mandelbrot set?

How does the main cardioid appear in the Mandelbrot set? I also wonder why something "weird" happens at a point with coordinate 2 on the actual coordinate line, I mean why is the point a sort of ...
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1answer
36 views

Is the basin of attraction to infinity open, and is it contained in the Fatou set?

I have managed to show that the interior of the basin of attraction to infinity, $A(\infty)$, is contained in the Fatou set so in order to show that $A(\infty)$ is in the Fatou set, I need to show ...
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1answer
40 views

Proving that all real numbers lie in Fatou set of $f(z)=\sin(z)/3$

The problem is this: Define the entire function $f$ by $$f(z)=\frac{\text{sin}(z)}{3}.$$ Show that all real numbers lie in the Fatou set $F_f$. Hint: For $x\in \mathbb{R}$ we have $|\text{sin}(x)|\...
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2answers
172 views

Algorithm for rendering non-filled Julia sets?

I have a commercial application, FractalWorks, for Mac OS. It creates 2D and 3D images of Mandelbrot and Julia sets (using complex numbers and the formula Zₓ₊₁ = Zₓ² + C.) It includes support for ...
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1answer
47 views

Relation between Filled Julia Set and Julia Set of a rational function

I am doing a small project in school, which is dedicated to exploring what shapes might Julia sets of rational functions take. However, as we've started investigating into the results we've got before ...
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1answer
56 views

Finding Julia set of $f(z)=z^2-1$

I am trying to determine the Julia set of $f(z)=z^2-1$. The fixed points are at $z=\frac{1\pm\sqrt{5}}{2}$ and I can see that for $|z|>\frac{1+\sqrt{5}}{2}$, then $|f(z)|>|z|$, so that $|f^n(z)|$...
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33 views

Applying Montel's Theorem to finding Fatou/Julia sets

I've learnt a few basic examples of finding the Julia and Fatou sets of non-linear entire functions e.g. $f(z)=z^2$, $f(z)=z^2-2$ using the iterative version of Montel's Theorem. I am struggling to ...
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2answers
72 views

How to prove that $\infty$ is an attractive fixed point for the function in Newton's method for the given function?

I'd like to ask to check for a solution to a homework problem, below. Define $\forall n \in \mathbb{N}, g_n(x):=\frac{x}{(1-x)^n}$. Let $F_n$ be the associated Newton function, i.e. $F_n(x):= x - \...
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62 views

Finding the fractal dimension of the Mandelbrot set using the box counting method

So I'm trying to calculate the fractal dimension of the perimeter of the mandelbrot set using the box-counting or Minkowski–Bouligand definition of fractal dimension. According to this definition, my ...
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1answer
52 views

Subsets of the Mandelbrot set

In order to define the Mandelbrot set $\mathbb M$, one looks at the sequences $$ s = f(0),(f\circ f)(0), (f\circ f \circ f)(0),\ldots, $$ where $f:\mathbb C \to \mathbb C,~z\mapsto z^2+c$ for ...
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0answers
73 views

Does the Mandelbrot set contains itself?

This question mentions that it contains similar shapes. But what I want to know, does the set contains a PERFECT shape of itself. Can I start to zoom and come to a state where I was already?
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38 views

Simple ways for checking if a point is not in a multibrot set

As is well-known, the Mandelbrot set (M-set for short) can be defined by considering the family of functions $f_c(z)=z^2+c$ for $c\in\mathbb{C}$, iterating them for each $c$ to obtain sequences $z_{n+...
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2answers
54 views

An heuristic description of parabolic points in rational iteration theory

I've been digging around the iteration of rational functions. By chance I came across a mapping $R(z)$ such that there is only one reppelling fixed point and two parabolic points. I'm still far from ...
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53 views

Proving the complete invariance of the Julia set

I’m doing an exercise that asks to prove that the Filled julia set $K_c$ and the Julia set $\partial K_c$ for $p_c(z) = z^2+c$ are completely invariant. I’ve shown that $K_c$ is completely invariant, ...
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1answer
46 views

Show that a Herman ring cannot occur for a polynomial

I would like to verify my proof of the following statement: Let $f(z)=z^n + O(z^{n-1})$ be a polynomial. Show that the Fatou set of $f$ does not contain a Herman ring. Proof: Suppose otherwise and ...
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0answers
32 views

Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ birational such that $fσ=σf$?

Consider the The standard quadratic involution.$ σ: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ which has the inverse $[z_1, z_2] → [z_1z_2^{-1},...
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1answer
81 views

Finding an upper bound to the order of finite subgroup of the automorphism group of rational map

Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map then we define $Aut(\phi)=\{f \in PGL_2(\Bbb C): f^{-1}\phi f(z)=\phi(z)\}$ Here, in general, the definition of a rational map is: Let $\mathbb{P}^...
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220 views

The mysterious numbers $ \frac{13}{20} $ and $20$?

Let $g(x) = x^6 - 30 x $ Let $h(x) = x^6 $ Let $f(x) = x^2 - 2 $ Let $r$ be a reduced fraction $0 < \frac{p}{q} < 2 $ with integers $p,q > 1$ Let $f_{n+1}(x) = f(f_n(x)) = f_n(f(x)) , ...
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2answers
72 views

Software for visualizing Julia sets

I would like a program where I can enter any complex function and see its Julia set. I have not been able to find a web program which does this. Most have functions which are of a fixed type, and you ...
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2answers
110 views

Why the extended complex plane $\mathbb{C_\infty} = \mathbb{C}\cup \{\infty\}$ is compact while $\mathbb{C}$ is not?

I think this is due to the fact the extended complex plane can be stereographically identified with the unit sphere in $\mathbb{R^3}$. However, an exact and proper explanation is required.
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1answer
85 views

Seeking a correct relationship between the number of zeros of an entire function and it's derivative.

As far as polynomials are concerned, we know by virtue of fundamental theorem of algebra that a polynomial of degree $n$ has exactly $n$ roots in $\mathbb{C}$ and it's derivative then will have $n-1$ ...
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1answer
57 views

Prove boundedness of a given entire function.

For a non-constant transcendental entire function $f$ and distinct constants $a$ and $b,$ suppose that $f(z)-az$ and $f(z)-bz$ are periodic with periods $x$ and $y$ respectively. I have to show ...
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1answer
33 views

Calculation of growth estimate for a certain transcendental entire function

Let $f$ be a transcendental entire function. Set $$h(z)= z+ f(z)/f'(z)$$ and $$ F(z) = (z-a) f(z)$$. Then prove that for some $k>0,$ $$T(r, h)\thicksim k T(r, F'/F)$$, where $T$ is the Nevanlinna'...
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0answers
39 views

Find the relationship between Nevanlinna's characteristics function for a certain transcendental entire function

For a certain transcendental entire function $f$, set $$h(z) = z + \frac{f(z)}{f'(z)} \quad \text{and} \quad F(z) = {(z-a)}{f(z)}.$$ Then for some $k> 0$, prove that $T(r,h) \thicksim k T(r, F'/F)...
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1answer
54 views

Calculating the integrated counting function for a certain transcendental entire function

For a transcendental entire function $f$, set $$h(z) = z + \frac{f(z)}{f'(z)} \quad \text{and} \quad F(z) = {(z-a)}{f(z)}.$$ Let $$E = \{p: f(p)=0\} \cup\{h(p): h'(p)=0\}$$ and suppose that $a\notin ...
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0answers
79 views

What is the general definition of thickness of a strange attractor?

In Chaosbook, at page 56, it is asked to find the thickness of Rössler strange attractor, by some means. However, up to this point, the book have only defined the thickness of an Henon map, as the ...
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1answer
39 views

For $p(x)= x (1/2-x)$ is the set of 0's of $p^n(x)$ dense in the Julia set?

I plotted the filled Julia set for $p(x) = -x^2+ x/2$ and the zero's of $p^n(x)$ for $n=10$, and the $1024$ $0$'s are mainly clustered around the boundary, as shown, where the filled Julia set, the ...
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0answers
94 views

Attracting and repelling fixed points and cycles

Consider the iteration which produces the Mandelbrot set: $f(z) = z^2+c$. At $c=0$, this iteration has an attractive fixed point. At $c=-1$, it has an attractive 2-cycle. As $c$ varies from $0$ to ...
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1answer
61 views

Show analytic isomorphism has no attracting fixed point

Let $f: D\mapsto D$ analytic isomorphism for connected open subset $D \subset \mathbb{C}$. How do we show that $f$ has no attracting fixed point, i.e. $|f'(z)| \ge 1$ when $f(z) = z$?
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1answer
99 views

Are all complex iterations in the form of $z_{n+1} = z_n + c$ fractals?

I've learned that many fractals can be described like the below: $$z_n \in F \text{ if } \limsup_{n\to\infty} |z_{n+1}| \leq i$$ Where $i$ is some number, usually I see 2, and $z$ is a member of the ...
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1answer
60 views

Identifying unordered set of complex data as originating from a particular Mandelbrot process?

Here is an old question which probably originated back in my high school days. When iterating the Mandelbrot $$z_{i+1} = {z_i}^2+c$$ we get a sequence of complex numbers. If we plot these in the ...
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1answer
46 views

How to calculate and simplify complex polynomials of higher order?

Background: I am back to an old hobby of mine: investigating the Mandelbrot set. This time I am doing it from a computational efficiency perspective. The Mandelbrot set can be defined through ...
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1answer
48 views

Injective holomorphic endomorphism whose image is the complement of a proper analytic subset is surjective.

Let $f: M \rightarrow M$ be an endomorphism of a connected complex manifold $M$. Assume that $f(M)$ is a dense open set in $M$ and that $M \setminus f(M)$ is an analytic subset of $M$ Question: Can I ...
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2answers
112 views

Julia set fractal generator created a Poincaré disk?

This happened to me when I was playing around on this site did I stumble upon a link between the Julia set and the geometry of a Poincaré disk? Does anyone know if there are documented occurrences of ...
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1answer
118 views

Complex dynamics: iterates of rational function : Julia and fatou set [closed]

Prove that attracting fixed points of rational map lies in fatou set and repelling fixed point of rational map lies in the julia set
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1answer
114 views

A quartic whose Julia set includes two quadratic Julia sets

The abstract for "Quartic Julia sets including any two copies of quadratic Julia sets" in Discrete & Continuous Dynamical Systems - A,36,4,2103,2112,2015-9-1, by Koh Katagata, states [...] for ...
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1answer
47 views

If $\phi(z)$ conjugates $f^{-1}(z)$ to $\phi(z)/\lambda$ then it also conjugates $f(z)$ to $\lambda\phi(z)$

On p.32 of Carleson/Gamelin Complex dynamics they show existence of a conjugation map for a repelling fixed point using the attracting case and I am trying convince myself of the statement "any map ...
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0answers
57 views

$\phi_{n+1}(z)-\phi_n(z)$ converges uniformly then $\phi_n(z)$ converges uniformly?

In Carleson/Gamelin Complex dynamics p. 31 it shown that an analytic function $f$ can be linearized near an attracting fixed point. Let $\phi_n(z)=\lambda^{-n}f^n(z)$. I have trouble understanding the ...
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0answers
85 views

Formal group law and Koenigs function conjecture !?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). $$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). $$ This equation has many solutions. ...

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